SECTION 8.2 TWO-DIMENSIONAL FIGURES - Cengagecollege.cengage.com/mathematics/bassarear/elementary_school/3e/... · SECTION 8.2 TWO-DIMENSIONAL FIGURES ... rhombus with one additional

508 CHAPTER 8 / Geometry as Shape

WHAT DO YOU THINK?

• In what ways are trianglesand quadrilateralsdifferent? In what ways arethey alike?

• How are circles andpolygons related?

• Can every polygon bebroken down intotriangles? Why or why not?

• Why do we use two wordsto name different triangles,but only one word to namedifferent quadrilaterals?

SECTION 8.2 TWO-DIMENSIONAL FIGURES

Think of geometric figures that people generally find pleasing, such as those inFigure 8.45. What words would you use to explain why these objects are inter-esting or appealing? When you look at the various objects and pictures, whatsimilarities do you see between certain objects and shapes—for example, tri-angles and hexagons?

When we discuss similarities and differences in my class, many geometricterms emerge in the discussion. Some students talk about similar shapes—forexample, hexagons in honeycombs and snowflakes, squares in some pictures,and triangles in others. Some students observe that many of the shapes aresymmetric. In explaining similarities, some students talk about the angles, thelength of sides, or the fact that some figures look similar. Many students ob-serve that even the more complex shapes can be seen as being constructed fromsimpler shapes, such as triangles and quadrilaterals.

(a) (b) (c)Carpenter’s Wheel quilt Honeycomb

FIGURE 8.45

b. On each of the Geoboards below, make a line perpendicular to the given line. Develop, describe, and justify a rule or pro-cedure for making sure that the two lines are perpendicular.

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(d)lamp post

FIGURE 8.45 (continued)

In a few moments, you will begin a systematic exploration of geometricshapes. Before you do, let us examine a very important framework for under-standing the development of children’s geometric thinking. This model was de-veloped by Pierre and Dina van Hiele-Geldorf in the late 1950s and is widelyused today. Essentially, the van Hieles found that there are levels, or stages, inthe development of a person’s understanding of geometry.2

Level 1: Reasoning by resemblance

At this level, the person’s descriptions of, and reasoning about, shapes isguided by the overall appearance of a shape and by everyday, nonmathemati-cal language. For example, “This is a square because it looks like one.” Studentsat this level may be made aware of the various properties of geometric objects (forexample, that a square has four equal sides), but such awareness can be overriddenby other factors. For example, if we turn a square on its side, the student may insistthat it is no longer a square but now is a diamond.

Level 2: Reasoning by attributes

At this level, the person can go beyond mere appearance and recognize and de-scribe shapes by their attributes. A student at this level, seeing the figure above,can easily classify it as a quadrilateral because it has four sides. However, a stu-dent at this level does not regularly look at relationships between figures. A stu-dent who argues that a figure “is not a rectangle because it is a square” isreasoning at this level.

Section 8.2 / Two-Dimensional Figures 509

(e)snowflake

(f)Native American basket weaves

Source: LeRoy Appleton, American Indian De-sign and Decoration (New York: Dover, 1950).

2My writing of this section has been informed by Thomas Fox’s “Implications of Research on Children’s Understanding of Geometry” in the May 2000 issue of Teaching Children Mathematics,pp. 572–576.

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Level 3: Reasoning by properties

At this level, the student sees the many attributes of shapes and the relation-ships between and among shapes. A student at this level can see that the squareand rhombus have many properties in common, such as opposite sides paral-lel, all four sides congruent, and diagonals that bisect each other and are per-pendicular. This enables the student to understand that a square is simply arhombus with one additional property—all the angles are right angles.

Level 4: Formal reasoning

Students at this level can understand and appreciate the need to be more sys-tematic in their thinking. When solving a problem or justifying their reasoning,they are able to focus on mathematical structures.

The investigations in the text and the accompanying explorations havebeen designed to be consistent with this approach. Accordingly, as you areworking in this chapter, reflect on your own thinking. Are you looking at theproblem only on a vague, general level? Are you fixated on just one attribute?Are you seeing relationships among triangles, among quadrilaterals? As youmove from what to why, are you able to move from solving a problem by ran-dom trial and error to being more systematic and careful in your approach?

With this model in mind, let us begin our exploration of shapes with an in-vestigation my students have found to be both fun and powerful. Even moreimportant than knowing the names of all these different shapes will be know-ing their properties and the relationships between and among the shapes. It is this knowledge that is used by artists, engineers, scientists, and all sorts ofother people.

INVESTIGATION

8.5Recreating Shapes from Memory

For this investigation, you will want to have a pencil and an eraser.

A. Look at Figure 8.46 for about 1 second. Then close the book and drawFigure 8.46 from memory.

Check the picture again for 1 second. If your drawing wasincomplete or inaccurate, change your drawing so that it is accurate.Check the picture again for 1 second. Keep doing this until your drawingis complete and accurate.

Now go back and try to describe your thinking processes as you triedto re-create the figure. From an information processing perspective, youreyes did not simply receive the image from the paper; your knowledge of geometry helped determine how you saw the picture. What did youhear yourself saying to help you remember the picture? Then read on. . . .

DISCUSSION

Some students see a diamond and 4 right triangles. Other students see a largesquare in which the midpoints of the sides have been connected to make a newsquare inside the first square. Yet other students see four right triangles thathave been connected by “flipping” or rotating them.


FIGURE 8.46

8.7

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B. Now look at Figure 8.47 for several seconds. Then close the book and tryto draw it from memory. As before, check the picture again for a fewseconds. If your drawing was incomplete or inaccurate, fix it. Keep doingthis until your drawing is complete and accurate. Then go back anddescribe the thinking processes you engaged in as you tried to re-createthe figure. . . .

DISCUSSION

This figure was more complex. Some students see a whole design and try to re-member it.

Some decompose the design into four black triangles and four rectangles asshown in Figure 8.48.

The figure can also be seen as being composed of 9 squares, which can alsobe seen in Figure 8.48. The four corner squares have been cut to make congru-ent right triangles. Each of the other four squares on the border has been cutinto two congruent rectangles.

Some students re-created this figure by seeing a whole square and thenlooking at what was cut out (see Figure 8.49). That is, they saw that they neededto cut out each corner, and they saw that they needed to cut out a rectangle on the middle of each side. Finally, they remembered to cut out a square in thecenter.

When I did this exercise myself, I saw a “square race track” with a similarpattern on each corner (Figure 8.50).

This is a famous quilting pattern called the Churn Dash. When peoplemade their own butter, the cream was poured into a pail called a churn. Thenthe churner rolled a special pole back and forth with his or her hands. At theend of the pole was a wooden piece called a dash, which was shaped like thefigure you saw. If you make many copies of this pattern and put them together,you can see what a Churn Dash quilt looks like.

There are several implications for teaching from this investigation. How aperson re-creates the figure is related to the person’s spatial-thinking prefer-ences and abilities. Different people “see” different objects. That is, not every-one sees the figure in the same way. Although there are differences in howpeople re-create the figure, very few people can re-create the figure withoutdoing some kind of decomposing—that is, without breaking the shape intosmaller parts. Being able to do this depends partly on spatial skills and partlyon being able to use various geometric ideas (congruent, triangle, square, rec-tangle) at least at an intuitive level.

Although some people manage to live happy, productive lives at the low-est van Hiele level, an understanding of basic geometric figures and the rela-tionships among them is often helpful in everyday life (for example, in homerepair projects and quilting) and in many occupations. Now that your interestin geometric figures has been piqued by this investigation and the pictures atthe beginning of the section, let us examine the characteristics and properties ofbasic geometric shapes.

Before we examine specific kinds of polygons, beginning with triangles, thefollowing investigation will serve to “open your thinking”—to get you to lookat polygons not only through the zoom lens, which reveals specific properties


FIGURE 8.47

FIGURE 8.48

FIGURE 8.49

FIGURE 8.50

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and definitions, but also through the wide-angle lens, in which you see all theattributes. For example, a person is not just a person. She might be a mother, asister, a daughter, a scientist, a Democrat, and so on. Should you become afriend of this person, you come to know her many facets. Similarly, if you be-come a “friend” of shapes, you come to know the many “sides” of the shapeswith which you are working.

INVESTIGATION

8.6All the Attributes

Look at the polygons in Figure 8.51. Think of all theattributes, all the characteristics,anything about the polygonsthat might be important to stateor measure. Write down yourlist before reading on. . . .

DISCUSSION

As you compare the following lists to yours, read actively. If you made thesame observation, did you use the same wording? If not, do you understandthe wording here? If you missed one of these attributes, why? Do you under-stand it now? Are there other attributes that you noticed?

A key idea here is to realize that there are lots of attributes and that know-ing these attributes and combinations of attributes of a shape helps chemistsand physicists to understand the behavior of a molecule or shape; helpsbuilders to know which shapes work together better either in terms of structureand strength or in terms of appearance; and helps artists and designers to makedesigns that are the most appealing. As you read on, think of multiple attrib-utes and of attributes that different objects have in common.


FIGURE 8.51

First figure Second figure

6 sides 6 sides

Top and bottom sides parallel to each other Top and bottom sides parallel to each other

All three pairs of opposite sides parallel

Concave Convex

2 sets of congruent sides 4 congruent sides; the other pair is also congruent

2 acute angles, 2 right angles, 2 reflex angles 4 obtuse angles, 2 right angles

3 pairs of congruent angles Opposite angles congruent

1 line of reflection symmetry No reflection symmetry

No rotation symmetry -turn rotation symmetry

Does not tessellate Tessellates

1�2

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INVESTIGATION

8.7Classifying Figures

Before we examine and classify important two-dimensional shapes, we firstneed to investigate the kinds of possible two-dimensional shapes. Althoughmost of elementary students’ exploration of two-dimensional figures willinvolve polygons and circles, it is important to know that these figuresrepresent just a small subset of the kinds of figures that mathematiciansstudy. Both circles and polygons are curves. A mathematical curve can bethought of as a set of points that you can trace without lifting your pen orpencil. If you watch young children making drawings, you discover that theymake all sorts of curves!

As you might expect, if we look at any set of curves, there are manyways to classify them. As I have done throughout this book, rather thangiving you the major classifications, I will engage you in some thinkingbefore presenting them. Look at the 13 shapes in Figure 8.52 and classifythem into two or more groups so that each group has a commoncharacteristic. Do this in as many different ways as you can, and then read on. . . .

FIGURE 8.52

DISCUSSION

One way to sort the figures is shown below. How would you describe the fig-ures in set A and the figures in set B? Do this before reading on. . . .

The figures in set A are said to be simple curves. We can describe simplecurves in the following way: A figure is a simple curve if we can trace the fig-ure in such a way that we never touch a point more than once. If you look atthe figures in set A, you can see that they all have this characteristic; and all thefigures in set B have at least one point where the pencil touches twice, no mat-ter how you trace the curve.

Set A

Set B

1 2 3 4 5 6 7

8 9 10 11 12 13


8.8

� Language �

What other words might youuse to describe theintersecting and notintersecting subsets?

Some students use thephrase “trace over,” and otherstalk about figures that “runover themselves” or “crossthemselves.” Other studentstalk about the set of figuresthat contain two smallerregions within each figure.

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Now look at the curves in sets C and D. How would you describe the fig-ures in set C and the figures in set D? Do this before reading on. . . .

The figures in set C are said to be closed curves. We can describe closedcurves in the following way: A figure is a closed curve if we can trace the fig-ure in such a way that our starting point and our ending point are the same. Ifyou look at the figures in set C, you can see that they all have this characteris-tic; and no matter how you try, you cannot trace the figures in set D with thesame starting and ending point.

Now look at the curves in sets E, F, and G. How would you describe the fig-ures in set E, the figures in set F, and the figures in set G? Do this before read-ing on. . . .

These three sets are interesting for two reasons. First, these sets are likely tobe generated in the classroom—both your classroom and the elementary class-room. Second, the language used to describe the three sets poses a challenge,for most people describe the figures in set E as consisting only of curvy lines,the figures in set F as consisting only of straight line segments, and the figurein set G as having both curvy and straight line segments. The challenge here isthat when mathematicians use the word curve, this word encompasses bothcurvy and straight line segments—a curve is a set of points that you can tracewithout lifting your pen or pencil. There is nothing wrong with students’ useof the terms curvy and straight. What is important is the realization that we areusing the words curve and curvy in different ways. We do this all the time ineveryday English. Recall the various uses of the word hot in Chapter 1: ”It sureis a hot day.” “I love Thai food because it is hot.” “This movie is really hot!”

Most of our investigations of curves will focus on simple closed curves.Looking at the descriptions above, try to define the term simple closed curve be-fore reading on. . . .

We will define a simple closed curve as a curve that we can trace withoutgoing over any point more than once while beginning and ending at the samepoint. The set of polygons is one small subset of the set of simple closed curves.

At this point, you might want to do the following activity with anotherstudent.

• Draw a simple closed curve.• Draw a simple open curve.

Set F

Set E

Set G

Set D

Set C


CLASSROOM CONNECTION

Using the terms developed inthis investigation, how wouldyou classify dot-to-dot pictures?

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• Draw a nonsimple closed curve.• Draw a nonsimple open curve.

Exchange figures with another student. Do you both agree that each of theother’s drawings matches the description? If so, move on. If not, take some timeto discuss your differences.

There is an important mathematical theorem known as the Jordan curve theorem, after Camille Jordan: Any simple closed curve partitions the plane intothree disjoint regions: the curve itself, the interior of the curve, and the exteriorof the curve. See the examples in Figure 8.53.

FIGURE 8.53

I know that many students’ reaction to this theorem is “Why do we need toprove something that is so obvious?” As mentioned before, being critical (“toexamine closely”) is an attitude that I invite. In the examples in Figure 8.53, de-ciding whether a point is inside or outside is easy. However, look at Figure 8.54.Although this figure is a simple closed curve, it is a rather complicated fig-ure, and such complicated shapes are encountered in some fields of science. Ispoint A inside or outside? How would you determine this? Think before read-ing on. . . .

FIGURE 8.54

Some wag once remarked that mathematicians are among the laziest people on earth because they are always looking for shortcuts and simplerways to solve problems. Thus you may be wondering whether someone has

A

ExteriorInterior

Exterior

Interior

(a) (b)


� History �

Generally, when a theorem is named after a person, it is named after the person who first proved the theorem.In this case, Jordan’s proofwas found to be incorrect, butthe theorem is still namedafter him!

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found an easier way to solve these problems. Indeed, someone has. I can usethe shape in Figure 8.55 to illustrate the method. Start at a point that is clearlyoutside the shape and draw a line segment connecting that point to the pointyou are looking at; it helps if you pick an outside point so that the line segmentwill cross the curve in as few points as possible. Each time you cross a point,it’s like a gate— if you were outside, you are now inside; if you were inside, youare now outside. Thus it is a relatively simple matter to determine that point Bis inside the curve.

FIGURE 8.55

Now go back to the first shape to see whether point A is inside or outside.As an active reader you will want to work on the figure to verify to your satis-faction that the method just described really does work.

Polygons

We are now ready to begin our exploration of polygons, which can be definedas simple closed curves composed only of line segments. Thus the simpleclosed curve in Figure 8.53(a) is not a polygon, whereas the simple closed curvein Figure 8.53(b) (which looks like the state of Nevada) is a polygon. On anypolygon, the point at which two sides meet is called a vertex, the plural ofwhich is vertices. The line segments that make up the polygon are called sides.

The word polygon has Greek origins: poly-, meaning “many,” and -gon,meaning “sides.” You are already familiar with many kinds of polygons. Just aswe found in Chapter 2 that the names we give numbers have an interesting his-tory, so do the names we give to polygons. The most basic naming classificationinvolves the number of sides (see Table 8.2).

Now that we have a good general definition of the term polygon, we canspend time examining triangles, quadrilaterals, and a few specific kinds ofpolygons.

Triangles

Triangles are found in every aspect of our lives— in buildings, in art, in science(see Figures 8.56 and 8.57). They are truly “building-block” shapes. Every bi-cycle I have seen has triangles. Bridges will always contain triangles. If you

B



The active and curious readerhere might realize that thisprocedure might haveapplications for solving mazes,and there is a branch ofmathematics that does analyzemazes. Another active andcurious reader might realizeand/or remember that mostchildren enjoy the challenge ofsolving mazes.

TABLE 8.2

Numberof sides Name

3 sides Triangle

4 sides Quadrilateral

5 sides Pentagon

6 sides Hexagon

7 sides Heptagon

8 sides Octagon

n sides n-gon

8.9

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look at the skeleton of buildings, and the scaffolding around the building, youwill always see triangles. Why? Rather than give you the answer, we will usethe next investigation to think about this question.

FIGURE 8.56 FIGURE 8.57

INVESTIGATION

8.8Why Triangles Are So Important

Cut some strips of paper from a file folder or other stiff material. Punch ahole in the ends and use paper fasteners (improvise if you need to; forinstance, you can use paper clips). Make one triangle and one quadrilateral,as shown in Figure 8.58. It need not be an equilateral triangle or a square.What do you see? . . .

FIGURE 8.58

DISCUSSION

As you saw, the triangle won’t move—we call it a rigid structure. However, thequadrilateral does move; it is not stable. Make another strip and connect twononadjacent vertices of your quadrilateral. What happens now? It will remainin the shape. If it is a square, it will remain a square; if it is a parallelogram, itwill remain a parallelogram. This is because the addition of that diagonal actu-ally created two triangles, which, as you have found, are rigid structures. Thenext time you walk about campus and about town, look for triangles. You willsuddenly see them everywhere!

As you already know, there are many kinds of triangles. A crucial goal ofthe next investigation is for your own understanding of triangles to becomemore powerful. Thus, as always, please do the investigation, rather than justreading through it quickly because “I already know this stuff.”


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INVESTIGATION

8.9Classifying Triangles

You will find nine triangles in Figure 8.59. Copy them and cut them out, andthen separate them into two or more subsets so that the members of eachsubset share a characteristic in common. Can you come up with a name foreach subset? What names might children give to different subsets? Do this inas many different ways as you can before reading on. . . .

FIGURE 8.59

DISCUSSION

STRATEGY 1: Consider sides

One way to classify triangles is by the length of their sides: all three sides hav-ing equal length, two sides having equal length, no sides having equal length.There are special names for these three kinds of triangles.

• If all three sides have the same length, then we say that the triangle is equilateral.

• If at least two sides have the same length, then we say that the triangle isisosceles.

• If all three sides have different lengths— that is, no two sides have the samelength— then we say that the triangle is scalene.

Which of the triangles in Figure 8.59 are scalene? Which are isosceles?Which are equilateral?

STRATEGY 2: Consider angles

We can also classify triangles by the relative size of the angles— that is, whetherthey are right, acute, or obtuse angles. This leads to three kinds of triangles:right triangles, obtuse triangles, and acute triangles.

• We define a right triangle as a triangle that has one right angle. • We define an obtuse triangle as a triangle that has one obtuse angle. • We define an acute triangle as a triangle that has three acute angles.

Many students see a pattern: A right triangle has one right angle, an obtusetriangle has one obtuse angle, yet an acute triangle has three acute angles. Whatwas the pattern? Why doesn’t it hold? Think before reading on. . . .


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The key to this comes from looking at the triangles from a different per-spective: Every right triangle has exactly two acute angles, and every obtuse tri-angle has exactly two acute angles; thus a triangle having more than two acuteangles will be a different kind of triangle. This perspective is represented inTable 8.3. Does it help you to understand better the three definitions given above?

STRATEGY 3: Consider angles and sides

This naming of triangles goes even further. What name would you give to thetriangle in Figure 8.60?

This triangle is both a right triangle and an isosceles triangle, and thus it iscalled a right isosceles triangle or an isosceles right triangle. How many pos-sible combinations are there, using both classification systems? Work on thisbefore reading on. . . .

There are many strategies for answering this question. First of all, we findthat there are nine possible combinations (see Figure 8.61). We can use the ideaof Cartesian product to determine all nine. That is, if set S represents trianglesclassified by side, {Equilateral, Isosceles, Scalene}, and set A represents tri-angles classified by angle, {Acute, Right, Obtuse}, then representsthe nine possible combinations.

FIGURE 8.61

However, not all nine combinations are possible. For example, any equilat-eral triangle must also be an acute triangle. (Why is this?) Therefore, “equilat-eral acute” is a redundant combination. However, it is possible to have scalenetriangles that are acute, right, and obtuse. Similarly, we can have isosceles tri-angles that are acute, right, and obtuse.

Name the two triangles in Figure 8.62. Then read on. . . .

FIGURE 8.62

Equilateral

Isosceles

Scalene

Acute

Right

Obtuse

S � AA �S �


TABLE 8.3

First angle Second angle Third angle Name of triangle

Acute Acute Right Right triangle

Acute Acute Obtuse Obtuse triangle

Acute Acute Acute Acute triangle

FIGURE 8.60

� Mathematics �

Did you notice the geometricbalance in Figure 8.61, whichrepresents the Cartesianproduct of the two sets oftriangles?


Some students have told methat this investigation is hugefor them, because they hadnever thought aboutorganization and classificationwith respect to triangles.Triangles just were!

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Both are obtuse, isosceles triangles. The orientation on the left is the stan-dard orientation for isosceles triangles. Any time we position a triangle so thatone side is horizontal, we call that side the base. As stated at the beginning ofthis chapter, students often see only one aspect of a triangle; for example, theysee the triangle at the left as isosceles but not also obtuse, and they see the tri-angle at the right as obtuse but not also isosceles.

As mentioned earlier, one of the new NCTM process standards is represen-tation. As I have also mentioned earlier, one definition of understanding has todo with the quantity and quality of connections the learner can make withinand between various ideas. In this case, a new representation of the relation-ship between and among triangles will help to deepen your understanding.More of these problems can be found in Exploration 8.13. These diagrams arealso being used more and more in elementary schools because of their tremen-dous potential to help children see more relationships and connections.

INVESTIGATION

8.10Triangles and Venn Diagrams

Recall our work with Venn diagrams in Chapter 2. Venn diagrams can helpus to understand how concepts are and are not related. Let us take twokinds of triangles: right triangles and isosceles triangles. Draw several righttriangles and draw several isosceles triangles. Which of these triangles couldbe placed in both categories? That is, which are right triangles and also



Children’s development of triangles is fascinating. In several studies, children weregiven many shapes and asked to identify the shapes. Young children tend toidentify the equilateral triangle in the “standard” position (base parallel to thebottom of the page) as a “true” triangle. They will often reject triangles such as theones below because they are too pointy or turned upside down. Recall level 1 inthe van Hiele model. One of my all-time favorite examples occurred when a first-grader was given the pattern shown in Figure 8.63 and was asked to continue the

FIGURE 8.63

pattern. After studying the pattern, she said, “Triangle, triangle, wrong triangle,triangle, triangle, wrong triangle, triangle. . . . The next shape is a right triangle!”3 Itis important to note that this child was doing wonderful thinking—she was seeingpatterns, and she was looking at attributes, and she was at the beginning of herunderstanding of triangles.

3From “Shape Up!” by Christine Oberdorf and Jennifer Taylor-Cox, published in the February 1999issue of Teaching Children Mathematics, pp. 340–345.

8.10

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isosceles triangles? Make a Venn diagram that illustrates the relationshipsbetween the right and isosceles triangles, and place each triangle in theappropriate region of the diagram.

Now draw several acute triangles and draw several equilateral triangles.Which of these triangles could be placed in both categories? How are thesetriangles related to each other? Make a Venn diagram that illustrates therelationships between the acute and equilateral triangles, and place eachtriangle in the appropriate region of the diagram.

DISCUSSION

When we have two groups of objects and look at how they are related, there arethree possible relationships. They may be disjoint—each object is in one or theother group; there may be overlap—some objects are in both groups; or onegroup may be a subset of the other. Each of these relationships (remember thevan Hiele levels) is represented with a different diagram, as shown in Figure 8.64.

FIGURE 8.64

Look at your Venn diagrams and think of what you just read. Do you wantto change your diagrams? If you couldn’t make the Venn diagrams because youjust didn’t understand the question, can you do so now? Do this before readingon. . . .

Because some triangles are both right and isosceles, those triangles can beplaced in the center, showing that they belong to both sets (see Figure 8.65). Be-cause equilateral triangles can contain only acute angles, all equilateral tri-angles are acute triangles. Hence the equilateral triangles are in a ring that isinside the ring that represents acute triangles.

FIGURE 8.65

Equilateral ∆

Acute ∆Isosceles ∆Right ∆


� Mathematics �

Interestingly, all right isoscelestriangles have the same shape.That is, if you found the ratioof lengths of the sides of tworight isosceles triangles,converted this to a percent,placed the larger one on acopy machine, and enteredthat percent in the reductionbutton, the smaller trianglewould come out. We willexplore this notion ofsimilarity in Section 9.3.

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Triangle Properties

You probably remember that the sum of the measures of the angles in any triangle is 180 degrees. How could you convince someone who does not know that?

This is a case in which the van Hiele levels are instructive. A level 2 activ-ity would be to have the students measure and add the angles in several tri-angles. If their measuring was relatively accurate, one or more students wouldsee the pattern and offer a hypothesis that the sum is always 180.

An example of a level 3 activity, which I recommend if you have neverdone it, is to cut off the three corners of a triangle as in Figure 8.66 and then putthe three corners together. What do you see?

An example of a level 4 activity would be a formal proof.

Special Line Segments in Triangles

There are four special line segments that have enjoyed tremendous influence in Euclidean geometry: angle bisector, median, altitude, and perpendicular bisector.

A median is a line segment that connects a vertex to the midpoint of the op-posite side.

In triangle SAT (Figure 8.67), is a median. Hence

FIGURE 8.67

An angle bisector is a line segment that bisects an angle of a triangle.In triangle ABC (Figure 8.68), is an angle bisector. Hence,

FIGURE 8.68

A perpendicular bisector is a line that goes through the midpoint of a sideand is perpendicular to that side. In triangle PEN (Figure 8.69), is a per-MX

i

A

B

D

C

m�DAC.m�BAD �AD

R AS

T

SR � RA.TR


FIGURE 8.66

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pendicular bisector of side because M is the midpoint of and is per-pendicular to

FIGURE 8.69

An altitude is a perpendicular line segment that connects a vertex to theside opposite that vertex. In some cases, as in in Figure 8.70, we need toextend the opposite side to construct the altitude.

In triangle ABC (Figure 8.70), is an altitude. Hence,

In triangle STA, is an altitude. Hence,

FIGURE 8.70

Many students have trouble with the idea of an altitude being outside thetriangle. This happens when we have an obtuse triangle oriented with one sideof the obtuse angle as the base of the triangle. If you are having trouble con-necting the definition of altitude in these situations, I recommend the following:Trace the triangle and cut it out. Stand it up so that is on the plane of yourdesk and T is above that plane. Now draw a line from T that goes “straightdown.” What do you notice?

If triangle STA were large enough so that you could stand with your headat point T, the length of line segment would tell you how tall you were!

After reading the preceding discussion, a skeptical reader might ask, “Sowhat? What is the value of these line segments, other than to mathematicians?”One response is that there is practical value, and another response is that thereare some really cool relationships here. Let me explain.

The segment with the most practical value is the median. The point whereall three medians meet is called the centroid and is the center of gravity of thetriangle (see Figure 8.71). That is, if you found this point, made a copy of thetriangle with wood, and placed the triangle on a nail at that point, the trianglewould balance. Center of gravity is related to balance and is an important con-cept in the design of many objects—cars, furniture, and art, to mention just a few.

The point where the three perpendicular bisectors meet is called the cir-cumcenter (see Figure 8.72). It turns out that this point is equidistant from eachof the three vertices of the triangle. Thus you can draw a circle so that all threevertices of the triangle lie on the circle and the rest of the triangle is inside thecircle. We say that the circle circumscribes the triangle.

TP

SA

B

AF

C P

T

SA

m�TPA � 90�.TPm�BFC � 90�.

m�BFA �BF

�SAT

E

PM

N

X

PN.MXPNPN,


FIGURE 8.71

FIGURE 8.72

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The point where the three angle bisectors meet is called the incenter (seeFigure 8.73). It turns out that this point is equidistant from each of the threesides of the triangle. Thus you can draw a circle so that the circle touches (is tan-gent to) each of the three sides of the triangle; in this case, the circle is inside thetriangle. We say that the circle is inscribed in the triangle.

The point where the three altitudes meet is called the orthocenter (see Fig-ure 8.74). The orthocenter has connections to other geometric ideas. For ex-ample, there is a connection between the construction of a parabola and theorthocenter of a triangle.

There are two amazing things about these line segments. The first is that inany triangle there are three medians, three perpendicular bisectors, three anglebisectors, and three altitudes. In each case, the three line segments will alwaysmeet at a single point— they are concurrent. That is, the three medians will al-ways meet at one point in any triangle, the three perpendicular bisectors willalways meet at one point in any triangle, and so on. Now, except for the case ofthe equilateral triangle, these points are not the same point. In fact, in a scalenetriangle, the four points are all different. However (and this is the other coolthing), there is a relationship among the centroid, circumcenter, and the ortho-center. In any triangle, they are collinear! Do you understand this? Figure 8.75illustrates this fact, which Leonard Euler first discovered. We even call the linecontaining these three points an Euler line. If you are curious, you can type inthese terms on a search engine. You will even find websites that have interac-tive features allowing you to see the situations where these four points are dis-tinct but collinear and the situations where they converge into one point.

FIGURE 8.75

Congruence

As you have seen from your geometry explorations, questions sometimes ariseabout whether two figures are “the same” or not. Such observations and ques-tions deal with the idea of congruence.

At an informal level, we can say that two figures are congruent if and onlyif they have the same shape and size. An informal test of congruence is to seewhether you can superimpose one figure on top of the other. This is closely con-nected to how children initially encounter the concept and is related to the dic-tionary definition: “coinciding exactly when superimposed.”4 That is, if onefigure can be superimposed over another so that it fits perfectly, then the twofigures are congruent.

Formally, we say that two polygons are congruent if and only if all pairs ofcorresponding parts are congruent. In other words, in order for us to concludethat two polygons are congruent, two conditions have to be met: (1) Each cor-responding pair of angles have the same measure and (2) each correspondingpair of sides have the same length. We use the symbol to denote congruence.

For example, in Figure 8.76, triangle CAT and triangle DOG are congruentiff and TC � GD.AT � OG,CA � DO,�T � �G,�A � �O,�C � �D,

�

A - centroid

B - circumcenter

C - orthocenter

AB

C


FIGURE 8.73

FIGURE 8.74

8.11

4Copyright © 1996 by Houghton Mifflin Company. Reproduced by permission from The AmericanHeritage Dictionary of the English Language, Third Edition.

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FIGURE 8.76

The notions of congruent and equal are related concepts. We use the termcongruence when referring to having the same shape, and we use the term equalwhen referring to having the same numerical value. Thus we do not say thattwo triangles are equal; we say that they are congruent. Similarly, when welook at line segments and angles of polygons, we speak of congruent line seg-ments and congruent angles. However, when we look at the numerical value ofthe line segments and angles, we say that the lengths of two line segments areequal and that the measures of two angles are equal.

Congruence is a big idea, both in geometry and beyond the walls of theclassroom. Many important properties and relationships come from exploringcongruence. The following investigations (and the related explorations) willhelp move your understanding of congruence to higher van Hiele levels, fromthe “can fit on top,” geometric reasoning by resemblance, to understanding thatall corresponding parts are congruent, to geometric reasoning by attributes, togeometric reasoning by properties, which we shall examine now.

INVESTIGATION

8.11Congruence with Triangles

For this you need a protractor and a compass, although you can improvisewithout them. Do both questions on a blank sheet of paper. First, see howmany different triangles you can make that have the following attributes:The base is 50 millimeters (mm). The angle coming from the left side of thebase is 30°, and the side coming from the left side of the base is 30 mm.Second, see how many different triangles you can make that have thefollowing attributes: The base is 30 mm. The angle coming from the left sideof the base is 30°, and the side coming from the right side of the base is25 mm. Do this before reading on. . . .

C

A

T D

O

G


� Outside the Classroom �

When do we need congruence in everyday life or in work situations? Take a fewminutes to think about this before reading on. . . .

Congruence is important in manufacturing; for example, the success ofassembly-line production depends on being able to produce parts that arecongruent. Henry Ford changed our world by conceiving of making cars not oneat a time but as many sets of congruent parts. For example, the left front fender ofa 2003 Dodge Caravan is congruent to the left front fender of any other 2003Dodge Caravan. One of the differences between a decent quilt and an excellentone is being able to ensure that all the squares are congruent. This is quitedifficult when using complex designs. Most of the manipulatives teachers use withschoolchildren (Pattern Blocks, unifix cubes, Cuisenaire rods, and fraction bars)have congruent sets of pieces.

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DISCUSSION

There is only one triangle that can be drawn in the first case. However, in thesecond case, there are two possible triangles. Thus there is not enough infor-mation about the triangle to specify exactly one triangle (see Figure 8.77).

FIGURE 8.77

In Section 8.1, we said that two points determine a line; here, we are lookingat what determines a triangle. This notion of determining or specifying is impor-tant in mathematics, both for congruence (When are two figures congruent?)and for definitions (How much do we need to specify to determine a shape?)For example, we can define a rectangle as a quadrilateral with four equal angles. Now a rectangle has many more properties than four congruent angles.However, mathematicians have discovered that this information—quadrilat-eral, four congruent angles— is sufficient so that only rectangles can be drawnthat meet that criteria. Thus the notion of “determines” is an important one in mathematics. In elementary school we do not get terribly technical, but thatis not the same as saying we just have fun and play around. When we ask children to explore well-focused questions, their understanding of shapes andrelationships between and among shapes and their ability to see and applyproperties can grow tremendously. When this happens, high school mathemat-ics makes much more sense!

Quadrilaterals

We found that we could describe different kinds of triangles by looking at theirangles or by looking at relationships among their sides. With quadrilaterals,which have one more side, new possibilities for categorization emerge: parallelsides, adjacent vs. opposite sides, relationships between diagonals, and the no-tion of concave and convex. Thus, how we go about naming and classifyingquadrilaterals is not the same as how we name and classify triangles.

In this book, we will define the following kinds of quadrilaterals:

• A trapezoid (Figure 8.78) is definedas a quadrilateral with at least onepair of parallel sides.

• A parallelogram (Figure 8.79) is defined as a quadrilateral in whichboth pairs of opposite sides are parallel.

• A kite (Figure 8.80) is defined as aquadrilateral in which two pairs ofadjacent sides are congruent.

50 mm A B

30 mm

30° 30°

C

D


� Language �

Some books define a trapezoidas having exactly one pair ofparallel sides. I have spokenwith many mathematiciansabout this particulardefinition—some prefer thedefinition used here and someprefer the other one. There isno right answer, but ratherreasons behind each one.There are two reasons why Iuse the definition given here.First, I would find it odd todefine an isosceles triangle ashaving at least two congruentsides but then a trapezoid ashaving exactly one set ofparallel sides. If we are goingto say a trapezoid has exactlyone pair of parallel sides, thenan isosceles triangle shouldhave exactly two congruentsides. There are reasons whyisosceles is defined the way itis, reasons beyond the scopeof this book, but I ampersuaded by parallel usagehere. Second, definingtrapezoid this way enables usto represent relationshipsamong quadrilaterals in amore elegant way.

FIGURE 8.78

FIGURE 8.79

FIGURE 8.80

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• A rhombus (Figure 8.81) is definedas a quadrilateral in which all sidesare congruent.

• A rectangle (Figure 8.82) is definedas a quadrilateral in which all angles are congruent.

• A square (Figure 8.83) is defined asa quadrilateral in which all foursides are congruent and all four angles are congruent.

An active reader may have noted that there are many other possible cate-gories of quadrilaterals. For example, there are many different quadrilateralsthat have at least one right angle or exactly three congruent sides. Just as wecalled the triangle with no sides congruent a scalene triangle, we could call aquadrilateral with no sides congruent a scalene quadrilateral. If we have namesfor quadrilaterals with two pairs of adjacent sides congruent and quadrilater-als with two pair of opposite sides congruent, why stop there? The primary rea-son why we have names for the quadrilaterals described above is that these arethe sets that mathematicians have found interesting, and specifying othergroups didn’t lead to anything beyond those groups; it just stopped, like a roadthat went nowhere. For example, we could define a rightquad as a quadrilat-eral with at least one right angle, and there are many different-looking quadri-laterals that would be rightquads. However, there is no name for this set ofquadrilaterals because examining them as a class wasn’t found to be useful.That is, it didn’t lead to practical applications or to other discoveries.

Diagonals One characteristic of all polygons with more than three sides isthat they have diagonals. The more sides in the polygon, the more diagonals.This term is probably familiar to most readers. However, before reading thedefinition of diagonal below, stop and try to define the term yourself so that it works for all polygons, not just squares and other quadrilaterals. Then readon. . . .

A diagonal is a line segment that joins two nonadjacent vertices in a polygon.

Figure 8.84 shows two different diagonals. One of the exercises will ask youto find patterns to determine the number of diagonals in any polygon.

FIGURE 8.84

Angles in quadrilaterals We know that the sum of the measures of the angles of any triangle is 180 degrees. Will the sum of the measures of thefour angles of any quadrilateral also be equal to one number, or will there be


FIGURE 8.81

FIGURE 8.82

FIGURE 8.83

� Outside theClassroom �

How many of each kind ofquadrilateral can you find ineveryday life?

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different numbers for different quadrilaterals? What do you think? What couldyou do to check your hypothesis? Once you believe that your hypothesis istrue, how could you prove it? Think before reading on. . . .

It turns out that for any quadrilateral, the sum of the measures of the fourangles is 360 degrees. The following discussion is an informal presentation ofone proof. If we draw a generic quadrilateral QUAD and one diagonal, as inFigure 8.85, what do you notice that might be related to this proof?

If you see two triangles and connect this to your knowledge that the sum of the measures of the angles of a triangle is 180 degrees, you have the key tothe proof. That is, you can conclude that the sum of all six angles must be360 degrees. However, these six angles are equivalent to the four angles of thequadrilateral!

INVESTIGATION

8.12Quadrilaterals and Attributes

Look at the three quadrilaterals below. Think about their various attributes.Recall Investigation 8.6. Now answer the following question: Which two ofthe quadrilaterals in Figure 8.86 are most alike and why? Do your thinkingand write your response before reading on. . . .

FIGURE 8.86

DISCUSSION

It is questions like this that help children, and adults, to move up the van Hielelevels. A good case can be made for different answers. Let us begin by simplylisting various attributes.

1 pair of sides 2 pairs of sides 2 pairs of sides1 pair parallel sides 0 parallel sides 0 parallel sides2 right angles 1 right angle 0 right angles1 obtuse angle 2 obtuse angles 0 obtuse angles1 acute angle 1 acute angle 3 acute angles0 reflex angles 0 reflex angles 1 reflex angleconvex convex concave

This list reflects various attributes on which we focused: congruence, par-allel, angles, and shape (convex and concave, which will be formally definedsoon). On the one hand, the two figures at the right are both kites and therefore“belong” together under that name. On the other hand, the two figures on theleft are both convex and both have right angles, so there is much in commonbetween the two of them also.

One of the most puzzling aspects of how mathematics has generally beentaught is how much of it is simply learning and reciting facts and theorems thatother people have learned. We ask our students to study mathematics, but werarely let them do mathematics. If we were to teach art this way, students would

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U

34 6

52

1Q

D

A

FIGURE 8.85

8.12

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learn techniques and be tested on how well they understood those techniques,but they would never get to do art. Most of the various groups advocatingchange in how mathematics is learned want to present students with problemswhere solving the problem means not simply applying an algorithm they havelearned but, rather, involves what we call mathematical thinking. I have soughtto give the activities in Explorations this flavor. It is more difficult to incorporatethis “doing mathematics” flavor into the textbook, but the discussion of mostinvestigations illustrates different solution paths to counter the widely heldmisconception that there is “one best way.” To a greater degree than many ofthe investigations in the text, the investigation that follows has this flavor ofdoing mathematics.

INVESTIGATION

8.13Challenges

This investigation brings together the notion of attributes and the notion ofdeterminism (e.g., two points determine a line), which are two of the bigideas of geometric thinking. How many different kinds of quadrilaterals canyou make that have exactly two adjacent right angles? Play around with thisfor a while on a piece of paper. Sketch different quadrilaterals that haveexactly two adjacent right angles. What do you see? Can you make anyconjectures? Can you prove them? Use whatever tools are available. As youdo, try to push yourself beyond random trial and error to being moresystematic, to thinking “What would happen if,” to looking at your solutionsto see what they have in common. . . .

DISCUSSION

Figure 8.87 shows three of many possibilities.

FIGURE 8.87

What do they all have in common besides two adjacent right angles? Thinkbefore reading on. . . .

They all have two parallel sides. That means they are trapezoids. A curiousreader might now be asking whether it is possible not to get a trapezoid. Whatdo you think? How might you proceed, rather than just using random trial anderror? Think before reading on. . . .

If you took high school geometry, you might be remembering a theoremthat said something like this: If two lines form supplementary interior angleson the same side of a transversal, then the lines are parallel. When we limit ourinvestigation to two adjacent right angles, we make two interior angles that arealso supplementary. Thus we know that the opposite sides of this quadrilateralmust be parallel. Hence the condition of adjacent right angles determines atrapezoid. Although we can vary the height of the figure and the lengths of theopposite sides, we can get only trapezoids. Figure 8.88 illustrates this.


FIGURE 8.88

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What if you make a quadrilateral where the two right angles are not adja-cent to each other, but opposite each other? Is it possible to have a concavequadrilateral with two right angles? These questions will be left as exercises.

A critical reader might be thinking this is just an exercise for students.Mathematicians don’t do this kind of stuff. But we do! A great deal of mathe-matics has been developed by mathematicians asking and exploring questionslike “I wonder whether this is possible?” and “I wonder what would happen if. . . .” In Chapter 9, I will talk more about a housewife who became intriguedby the question of under what conditions a pentagon would tessellate. Herwork was noticed by a mathematician who encouraged her, and the work shedid made a significant contribution to the field of tessellations. My belief is thatif you experience the doing of mathematics in this course, you will do this withyour children, who will then retain the love of numbers and shapes that theyalmost universally bring to kindergarten and first grade!

We used Venn diagrams to deepen our understanding of triangles in In-vestigation 8.10. We will do so again with quadrilaterals.

INVESTIGATION

8.14Relationships Among Quadrilaterals

Consider the Venn diagram and set of quadrilaterals shown in Figure 8.89.What attributes do all of the quadrilaterals in the left ring have? Whatattributes do all of the quadrilaterals in the right ring have? By the nature ofVenn diagrams, the quadrilaterals in the middle section have attributes ofboth right and left.

FIGURE 8.89

DISCUSSION

There are a number of ways to answer the question. Let us begin at the mostdescriptive level.

In the left ring, all of the shapes have four right angles, they all have oppo-site sides congruent, and they all have opposite sides parallel. Some studentsstate it slightly differently, saying that the shapes all have two pairs of congru-ent sides and two pairs of parallel sides.

In the right ring, all the shapes also have opposite sides congruent and op-posite sides parallel. Another attribute they possess is that all four sides arecongruent. The shape that is in the center, by definition, must have the attri-butes of both rings— four right angles and four congruent sides.

There is a name for figures that are in both rings—squares.There is a name for figures that are in the left ring—rectangles.There is a name for figures that are in the right ring—rhombi.


8.13


The act of classifying is one ofthe big ideas of mathematicsand is found in every branchand at every level ofinstruction. It begins withattribute blocks and materialswith preschool children.Materials such as buttons canbe sorted, using more than oneattribute, in many differentways (the shape of the button,the number of holes, the color,and so on). Plastic rings, largeenough to hold many objects,are used to create Venndiagrams.

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Using set language from Chapter 2, we say that squares are the intersectionof rectangles and rhombi. Some students will have noticed that all of theseshapes are parallelograms. Thus we could actually add another ring encirclingall the shapes in Figure 8.89. That is, all rectangles are parallelograms, allrhombi are parallelograms, and all rhombi are parallelograms.

Relationships Among Quadrilaterals

It turns out that we can view the set of quadrilaterals in much the same way we view a family tree showing the various ways in which individuals are re-lated to others. Figure 8.90 shows one of many ways to represent this familytree for quadrilaterals. Take a few moments to think about this diagram and toconnect it to what you know about these different kinds of quadrilaterals. Writea brief description. Does it make sense? Does it prompt new discoveries in yourmind?

FIGURE 8.90

One way to interpret this diagram is to say that any figure contains all ofthe properties and characteristics of the ones above it. The quadrilateral at thetop represents those quadrilaterals that have no equal sides, no equal angles,and no parallel sides; this is analogous to the scalene triangle. The kite and thetrapezoid represent two constraints that we can make: two pairs of congruent,adjacent sides or one pair of sides parallel. If we take a kite and require all foursides to be congruent, we have a rhombus. If we take a trapezoid and requireboth pairs of opposite sides to be parallel, we have a parallelogram. If we re-quire the angles in a parallelogram to be right angles, we have a rectangle. If werequire all four sides of a parallelogram to be congruent, we have a rhombus.Both the rhombus and the rectangle can be transformed into squares with onemodification—requiring the rhombus to have right angles or the rectangle tohave congruent sides. A key point is to begin to see connections and relation-ships among figures. Many students find, in this course, that their picture of

Kite

Morespecific

Moregeneral

Trapezoid

Quadrilateral

Square

Rectangle

Rhombus

Parallelogram


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geometry changes from looking like a list of definitions and properties to look-ing more like a network with connections among the various figures. Thisquadrilateral family tree can also help students to realize why mathematicsteachers say that a square is a rectangle and it is a rhombus: It has all the prop-erties of each! In everyday language, we say that a square is a special kind ofrhombus and a special kind of rectangle. In mathematical language, we say thatthe set of squares is a subset of the set of rhombuses and a subset of the set ofrectangles. The Venn diagram in Figure 8.91 illustrates this relationship.

Convex Polygons

Another concept that emerges with polygons having four or more sides is theidea of convex. Before reading on, look at the two sets of polygons in Fig-ure 8.92, convex and concave (not convex). Try to write a definition for convex.Then read on. . . .

FIGURE 8.92

For many students, defining convex is much like defining balls and strikesin baseball. As one umpire once said, “I knows it when I sees it.” Many studentsfocus on the word concave and say that the figure has at least one part that iscaved in. Such a description is acceptable at level 1 on the van Hiele model, butwe need a definition that is not as vague as “caved in.”

Examine the following definition to see whether it makes sense. Think be-fore reading on. . . .

A polygon is convex if and only if the line segment connecting any twopoints in the polygonal region lies entirely within the region.

If a polygon is not convex, it is called concave or nonconvex.

Looking at diagonals is an easy way to test for concave and convex. If any diagonal lies outside the region, then the polygon is concave. In poly-gon ABCDE in Figure 8.93, the diagonal AD lies outside the region.

Other Polygons

Although most of the polygons we encounter in everyday life are triangles andquadrilaterals, there are many kinds of polygons with more than four sides.Stop for a moment and think of examples, both natural and human-made ob-jects. Then read on. . . .

Convex

Concave


Rhombus Square Rectangle

4 ≅ angles4 ≅ sides

FIGURE 8.91

B

A

D

EC

FIGURE 8.93

� Language �

Your instructor might havehad you explore the definitionof convex in Exploration 8.9.You may be surprised to knowthat there are multiple ways toview this idea. A commonalternative to the definitiongiven at the right is that afigure is convex if and only if ithas no reflex angles. Firstmake sure you understand thedefinition. Do you see theequivalence of the twodefinitions? Do you like onebetter than the other? Onceagain, mathematics is not ascut and dried as many peoplebelieve!

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All of the figures in Figure 8.94 are polygons.

• The stop sign is an octagon—an eight-sided polygon.

• The common nut has a hexagonal shape—a six-sided polygon.

• The Pentagon in Washington has five sides.

Let us examine a few important aspects of polygons with more than four sides.

First, we distinguish between regular and nonregular polygons. What doyou think a regular pentagon or a regular hexagon is? How might we define it?Think about this and write down your thoughts before reading on. . . .

A regular polygon is one in which all sides have the same length and allinterior angles have the same measure.

What do we call a regular quadrilateral? What about a regular triangle?Is it possible for a regular polygon to be concave?Think about these questions before reading on. . . .

A regular quadrilateral is called a square. A regular triangle is called anequilateral triangle. A regular polygon cannot be concave.

A critical reader might be wondering whether you can have a polygon—let’s say a pentagon—where all the sides have the same length but not all theangles have the same measure. And what about the converse: Can you have apentagon where all the angles have the same measure but not all the sides havethe same length? What do you think? This will be left as an exercise.


FIGURE 8.94

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INVESTIGATION

8.15Sum of the Interior Angles of a Polygon

We found that the sum of the degrees of the interior angles of any triangle is180, and the sum for any quadrilateral is 360. What do you think is the sumof the interior angles of a pentagon? Can you explain your reasoning? Canyou find a pattern in this progression that will enable you to predict the sumof the interior angles of any polygon—for example, one with 10 sides orwith 100 sides? Work on this before reading on. . . .

DISCUSSION

From Table 8.4, many students can see that the sum increases by 180 each timebut cannot come up with the general case. The solution to this question comesfrom connecting the problem-solving tool of making a table to the seeing of patterns to seeing “increases by 180” as equivalent to “these are multiples of 180.” We can represent this equivalent representation in a fourth column that contains 180, and so on. What do you see now? Then readon. . . .

The number we multiply 180 by is 2 less than the number of sides in thepolygon. Therefore, the sum of the angles of a polygon having n sides will beequal to

The next step is to ask why. That is, this investigation takes an unprovedhypothesis that the sum will increase by 180 degrees each time we add a side.If that hypothesis is true, then it follows that the sum for any polygon will be

but why is this true? What do you think? That question will be leftas an exercise.

There are two kinds of angles that we will add to our consideration. Wehave spoken of interior angles; we can also speak of exterior angles. Draw whatyou think would be the exterior angles of the polygons in Figure 8.95. Thenread on. . . .

180�n � 2�,

�n � 2�180.

3 � 180,2 � 180,


TABLE 8.4

Sides Sum Reasoning

3 180

4 360 Increases by 180 each time

5 540

6 720

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FIGURE 8.95

This is one of those cases where mathematics is counterintuitive. An exte-rior angle is an angle formed by a side of the polygon and the extension of theside adjacent to that side. Now go back and see whether you can draw the ex-terior angles for the two polygons. How many exterior angles does a polygonhave? Then, and only then, look below. . . .

FIGURE 8.96

The diagrams in Figure 8.96 illustrate the exterior angles. As you may haveguessed, there is an exterior angle associated with each interior angle.

The sum of the interior angles of a polygon is What do youthink is the sum of the exterior angles of a polygon? . . .

Amazingly, the sum of the exterior angles of any polygon is 360 degrees.You can see the proof for yourself by placing a pencil on vertex C of the triangleabove. Now “walk” the pencil around the triangle. That is, move it to vertex O,now rotate the pencil so you are pointing toward W; note that the amount ofturn is exactly the exterior angle at O. Now walk the pencil to W. Turn the pen-cil at W so you are pointing toward C; again, the amount of turn is equal to theexterior angle at W. Finally walk to C and turn to face O. You find that the pen-cil has made one complete turn. Do the same with quadrilateral PONY. Onceagain, you find you make one complete turn.

All polygons have interior and exterior angles. When we focus on regularpolygons, there is an additional kind of angle. A central angle of a regular poly-gon consists of a vertex at the center of the polygon and the two sides connect-ing the center to adjacent vertices of the polygon (see Figure 8.97). The size ofthe central angle is not difficult to determine if you recall that one completeturn around a point is equal to 360 degrees. The measure of the central angle ofa regular n-gon (a polygon with n sides) is equal to This andother theorems developed in this section are used to advance our understand-ing of the structure of geometry, which is then used by people in various fields.For example, graphic artists use their sense of geometric structure to make verybeautiful and intricate designs, both on everyday objects like fabric and postersand as pieces of art.

360�n degrees.

180�n � 2�.

W

CO

Y

N

PO


� Mathematics �

If you thought all mathematicsproofs were like the two-column proofs in high schoolgeometry, you were mistaken.Many, many mathematicalproofs are fun and based oncommon sense that isaccessible to all people!

FIGURE 8.97

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Curved Figures

There is one more class of two-dimensional geometric figures that we need todiscuss: those figures that are composed of curves that are not line segments.

How many words do you know that describe such shapes? Think and thenread on. . . .

There are many such geometric figures— for example, circle, semicircle,spiral, parabola, ellipse, hyperbola, and crescent (see Figure 8.98).

FIGURE 8.98

In this course, we will focus on the simplest of all curved geometric figures:the circle. Stop for a moment to think about circles. How would you define acircle? Try to do so before reading on. . . .

A circle is the set of points in a plane that are all the same distance from agiven point, the center.

You are probably familiar with the following basic vocabulary for circles.In Figure 8.99, C is the center of the circle.The line segment is called a radius, the plural of which is radii.The line segment is called a diameter.The line segment is called a chord.The line which intersects the circle only at point D, is called a tangent.An arc is any part of a circle. We use two letters to denote an arc if the arc

is less than half of the circle. However, for larger arcs, we use three letters. Doyou see why?

We do this to distinguish between the arc at the left in Figure 8.100and the arc at the right because both of them have points A and D asendpoints.

FIGURE 8.100

On the basis of these descriptions and your previous experience with circles, try to write a definition for each of these terms. Then compare your defi-nitions with the ones below. . . .

A

D

B

Y

X

C

A

D

B

Y

X

C

�ABD�

�,�AD�

�

PQi

,XYABCA


A

C

B

X

Y

D

Q

P

FIGURE 8.99

304150_ch_08_02.qxd 1/16/04 6:07 AM Page 536

A radius of a circle is any line segment that connects a point on the circleto the center.

A diameter of a circle is any line segment that connects two points of thecircle and also goes through the center of the circle.

A chord is any line segment that connects two points of the circle. Thus, adiameter is also a chord.

A tangent line intersects a circle at exactly one point.An arc is a subset of a circle— that is, a part of a circle.

Coordinate Geometry

We have explored some of the territory of two-dimensional geometry. There isso much more! Most of our focus has been on polygons, especially triangles andquadrilaterals. We hope your understanding has moved to higher levels in thevan Hiele model. One of the themes of this book is multiplicity—multiple so-lution paths to most problems, multiple connections between and among ideas,and multiple ways of representing many ideas. One of the most powerful waysto illustrate the value of multiplicity is through coordinate geometry. Up to now, all of our exploration has been without coordinates— that is, we havesimply used sketches of geometric figures. Let us look at what is added whenwe place the figures in a coordinate plane.

A brief review of the Cartesian coordinate system

Any point on a plane can be represented by an ordered pair. The first numberrepresents the point’s horizontal distance from the center of the coordinateplane, which is called the origin and is denoted by the ordered pair Thesecond number represents the point’s vertical distance from the origin. At sometime during the development of mathematics, mathematicians adopted theconvention that right is positive, left is negative, up is positive, and down isnegative.

As we noted at the beginning of this chapter, a powerful contribution of theGreeks was to move us from the how to the why. In this chapter, we have ex-amined some geometric proofs, and you have explored proofs in Explorations.It turns out that some geometric proofs that are very difficult in “normal” rep-resentation are actually quite simple with coordinate geometry. Before we getto some proofs, let us do one investigation to review your skills with the coor-dinate system.

INVESTIGATION

8.16What Are My Coordinates?

First, let’s play a game. I’m thinking of a rectangle. Three of its coordinatesare and What is the fourth coordinate? Do this yourselfbefore reading on. . . .

�3, 10�.�7, 5�,�3, 5�,

�0, 0�.


� History �

For many hundreds of years,algebra and geometry weredeveloped for the most partseparately. It was in the 1600sthat René Descartes began themathematical work ofconnecting geometry toalgebra.

304150_ch_08_02.qxd 1/16/04 6:07 AM Page 537

DISCUSSION

Using their basic understanding of the properties ofrectangles and their intuitive, visual understanding ofthe coordinate plane (see Figure 8.101), most peoplewill deduce that the fourth coordinate is

Now, let us look at how to make this pattern evenmore obvious. Let us move the parallelogram so thatthe bottom left vertex is at the origin (see Figure 8.102).That is, we will move the whole rectangle 3 units to theleft and 5 units down. In Chapter 9, we will call this a

translation. Now what are the coordinates ofthe rectangle? Find this before reading on. . . .

With this strategic placement of the coordinates, the assertion that eitherthe x values or the y values are the same is even easier to see.

We need to develop one more idea before we jump more deeply into coor-dinate geometry, and that is to learn how to find the distance between twopoints on the coordinate plane. If you took geometry in high school, you prob-ably memorized this formula. However, if you read this carefully, you will findthat it doesn’t have to be memorized. Although it is not simple, it is not hard tounderstand.

INVESTIGATION

8.17Understanding the Distance Formula

Let us consider two random points on theplane: and If we con-sider the line segment connecting these two points to be the hypotenuse, we can draw the two sides of the triangle (see Figure 8.103).What must be the coordinates of the thirdvertex, E? Remember the discussions in thischapter about determinism—in this case,both the x and y values of the third vertex are determined. Think before reading on. . . .

DISCUSSION

The coordinates of that third vertex are Do you see why? If not, recallthat this vertex is the same horizontal distance from the origin as the top ver-tex; thus it has the same x value. Similarly, it is the same height above the ori-gin as the left vertex; thus it has the same y value.

Now we can apply the Pythagorean formula: This formulasimply says that in any right triangle, the length of the hypotenuse is equal tothe square root of the sum of the squares of the two sides.

If this seems a bit overwhelming, look at Figure 8.104 and connect it to Fig-ure 8.103. That is, in order to find the distance from Y to S, we have to find thedistance from Y to E and the distance from E to S. Then we will square thosedistances, add them, and take the square root.

But the distance from Y to E is easy because it’s a horizontal line. It is justthe difference between the x values— that is, Similarly, the distancefrom E to S is the difference between the y values— that is, y2 � y1.

x2 � x1.

c � �a2 � b2.

�x2, y1�.

S � �x2, y2�.Y � �x1, y1�

��3, �5�

�7, 10�.


(3, 10)

(3, 5) (7, 5)

FIGURE 8.101

(0, 5) (4, 5)

(0, 0) (4, 0)

FIGURE 8.102

S(x2, y2)

Y(x1, y1) E( , )

FIGURE 8.103

b

a

c

FIGURE 8.104

304150_ch_08_02.qxd 1/16/04 6:07 AM Page 538

Thus the distance from Y to S, substituting our distances into the Pytha-

gorean theorem must be Phrased in English, if we square the x distance and the y distance, add

them, and take the square root, we have the distance between any two points.Now we are ready for one of the proofs. Recall that a parallelogram is de-

fined as a quadrilateral in which opposite sides are parallel. Using only thisknowledge, we can prove that the opposite sides of a parallelogram must becongruent.

INVESTIGATION

8.18

The Opposite Sides of a ParallelogramAre Congruent

Let us begin by sketching aparallelogram on thecoordinate plane (see Figure 8.105). Since theopposite sides are paralleland since, from an algebraicperspective, parallel lineshave the same slope, weknow that the slopes of theopposite sides are equal.Thus, if we place one vertexat the origin and the nextvertex at we have one side lying on the x axis. Because we know thatPL and NA must be parallel, we can draw NA parallel to the x axis. Thus, ifwe let N be represented as the point we know that the y coordinate ofA must also be b. Do you see why?

DISCUSSION

If not, recall that slope is the ratio of rise to run. That is,

We know that the slope of PL is 0. Thus the slope of NA must also be zero. If both N and A have the same y coordinate, b, then the slope is

Because PLAN is a parallelogram, the other two sides must also be paral-lel— that is, they must have the same slope. Let us first determine the slope ofAL and PN.

The slope of PN is

The slope of AL is

What does this tell us about the relationship among a, d, and c?

It means that since the two lines have to have the same slope and thus beparallel, must be equal to a. That is, which is equivalent to

We can now substitute for d and have the new representation of thecoordinates of the points of the parallelogram (see Figure 8.106).

�a � c�d � a � c.

a � d � c,d � c

b � 0d � c

�b

d � c.

b � 0a � 0

�ba

.

�b � b��d � a� � 0.

y2 � y1

x2 � x1.

�a, b�,

�c, 0�,

��x2 � x1�2 � � y2 � y1�2.


N (a, b) A (d, b)

L (c, 0)P (0, 0)

FIGURE 8.105

304150_ch_08_02.qxd 1/16/04 6:07 AM Page 539

FIGURE 8.106

Next we must prove that the lengths of opposite sides are equal. Let’s beginwith the easier case, the horizontal sides. The distance from P to L is thatis, it is c. The distance from N to A is

What about the distance from P to N and from L to A? Using the distanceformula from Investigation 8.17, we can find the length of PN— that is, the dis-tance between P and N.

Now, we find the distance from L to A.

Because the two distances are the same, the lengths of the two line seg-ments are congruent. Therefore, we have proved that the opposite sides of aparallelogram are the same length.

INVESTIGATION

8.19Midpoints of Any Quadrilateral

Now let us investigate one of my favorite theorems in geometry. I want youto see it first; then we will prove it. On a blank piece of paper, draw aquadrilateral, any quadrilateral. Find the midpoints of each side. Irecommend using the metric side of your ruler. If you don’t have a ruler, youcan still find the midpoints. Do you see how? . . . Yes, you can fold the paperto find the midpoints of each side. Now make a new quadrilateral byconnecting the four midpoints consecutively. What do you see?

Do another one. This time, make a scalene quadrilateral—no congruentsides, no parallel sides. What you just observed will always happen. That is,in all cases, you will wind up with a parallelogram. Now, let’s prove that.

DISCUSSION

Before we can do the proof, we simply need to know how to find the midpointof a line segment. In this case, the theorem makes intuitive sense, and I will simply present it. The midpoint of the line segment connecting points

and shown in Figure 8.107, is That is, it is the aver-

age (mean) of the x values and of the y values.We will prove the theorem for all quadrilaterals. As we saw from the

quadrilateral hierarchy, once we prove a theorem for one quadrilateral, it is true

�x1 � x2

2,

y1 � y2

2 �.�x2, y2�,

�x1, y1�

LA � ��a � c� � c�2 � �b � 0�2 � �a2 � b2

PN � ��a � 0�2 � �b � 0�2 � �a2 � b2

�a � c� � a � c.c � 0;

N (a, b) A (a + c, b)

L (c, 0)P (0, 0)


(y2, y2)

(y1, y1)

FIGURE 8.107

304150_ch_08_02.qxd 1/16/04 6:07 AM Page 540

for all quadrilaterals below that quadrilateral. Because we are dealing with anyquadrilateral, we cannot assume any properties—congruence or parallel. Thuswe will set one vertex at the origin and one vertex on the x axis Thenthe other vertices are at arbitrary points and See Figure 8.108.

FIGURE 8.108

We will label the original quadrilateral HORS and the quadrilateral formedby the midpoints CANT.

Using the midpoint formulas, we determine the coordinates of the mid-points and then connect them. We now need to show that the opposite sidesmust be parallel— that is, that they have the same slope.

Let us begin with CA and TN.

A little algebra makes this next step much easier. If we multiply the top andbottom of this expression by 2, we don’t change the value.

Doing this, we have

Now for TN.

If we multiply the top and bottom of this expression by 2, we have

Thus we have shown that the slope of and that the slope ofThat is, these two line segments are parallel.

By a similar means, we can show that the slope of CT and the slope of ANare equal.

TN � c�b.CA � c�b

Slope of TN ��e � c� � e�d � b� � d

�cb

Slope of TN �

e � c2

�e2

b � d2

�d2

Slope of CA �c � 0

�a � b� � a�

cb

Slope of CA �

c2

� 0

a � b2

�a2

O(d, e)

R(b, c)

S(a, 0)

H(0, 0)

C( , 0)a

2

A( , )a + b2

c2

N( , )b + d2

e + c2

T( , )d2

e2

�d, e�.�b, c��a, 0�.�0, 0�


304150_ch_08_02.qxd 1/16/04 6:07 AM Page 541

Thus we have proved that if you take the midpoints of any quadrilateraland connect them in turn, you will always get a parallelogram. To prove thatby other means is much more tedious and difficult.

There is much more to coordinate geometry! We will revisit coordinategeometry in Chapter 9, and you will find coordinate geometry in the uppergrades of elementary school!


1. Write down all the attributes of each of the figures.

a. b. c.

2. In each case below, which two figures are most alike? Ex-plain your reasoning.

a.

b.

c.

3. Describe all the geometric shapes you see in the quilt de-signs below:

a. b.

Fool’s Puzzlec.

Diamonds in the Sky

Source: Jinny Beyes, The Quilter’s Album of Blocks & Borders(Delaplane, VA.: EPM Publications, Inc., 1986) p. 185.

4. Look at the shape at the right.

a. What do you see? (This is an open-ended question.)

b. Select several nonsimpleshapes. Name them andjustify each name. For ex-ample, are the quadrilater-als in each of the fourcorners kites or do theyjust look like kites?

5. Name at least six different polygons that you can see inthis shape. Trace and number each shape.

6. Try to make each figure on Geoboard Dot Paper and onIsometric Dot Paper. If you can make the figure, do soand explain why your figure is an example of the speci-fied triangle or quadrilateral. If you cannot make it, ex-plain why you think it is impossible to make the figureon that type of grid. Your reasoning needs to be based onproperties and attributes (see the van Hiele discussion),as opposed to, for example, “It’s a right triangle becauseit looks like a right triangle.”

a. acute scalene triangle

b. right isosceles triangle

c. obtuse isosceles triangle

d. equilateral triangle

e. trapezoid

f. kite

EXERCISES 8.2

304150_ch_08_02.qxd 1/16/04 6:07 AM Page 542

g. parallelogram

h. rectangle

i. rhombus

j. square

k. square with no sides parallel to the sides of the paper.

7. For each figure below, write “polygon” or “not a poly-gon.” If it is a polygon, also write “convex” or “concave.”

8. The definition of a regular polygon states that all sideshave the same length and all interior angles have the same measure. Why is the second part of the defini-tion necessary? That is, why can’t we just say that a poly-gon is a regular polygon if all the sides are the samelength?

9. How many different quadrilaterals can you make thathave at least one pair of adjacent congruent sides? Sketch and label your figures. For example, you can make manydifferent trapezoids, but they are all trapezoids. See thefigures below.

10. a. Is this figure a kite? Why or why not?

b. Is this figure a rectangle? Why or whynot?

c. Is this figure an isosceles triangle? Whyor why not?

11. Write directions for making the figures following. Fol-lowing your directions, the reader should be able tomake the same figure.

a. b. c.

d. e. f.

12. In each case below, determine whether the two figuresare congruent only by using your mind. That is, you can-not trace one figure and see whether it can be superim-posed on the other figure. Describe your reasoning—that is, how you arrived at your conclusion.

a. Below are two parallelograms made on a Geoboard.

b. Below are two figures made with tangram pieces.

c. Below are two pairs of hexominoes.

(1) (2)

13. Describe all quadrilaterals that have these characteris-tics. If there is more than one, say so.

a. A quadrilateral with opposite sides parallel

b. A quadrilateral with 4 right angles

c. A quadrilateral with all sides equal

d. A quadrilateral in which the diagonals bisect eachother

e. A quadrilateral in which the diagonals are congruent

f. A quadrilateral in which adjacent angles are congruent

g. A quadrilateral in which opposite angles are equal

h. A quadrilateral in which no sides are parallel

i. A quadrilateral with 4 congruent sides and 2 pairs ofcongruent angles

(d) (e)

(a) (b) (c)


304150_ch_08_02.qxd 1/16/04 6:07 AM Page 543

j. A quadrilateral with 4 congruent angles and 2 pairs ofcongruent sides

14. Draw each of the following or briefly explain why sucha figure is impossible.

a. An isosceles trapezoid

b. A concave quadrilateral

c. A curve that is simple and closed but not convex

d. A nonsimple closed curve

e. A concave equilateral hexagon

f. A concave pentagon having three collinear vertices

g. A pentagon that has 3 right angles and 1 acute angle

15. In each case, explain and justify your answer— that is, indicate why you think there are none, just one, ormany.

a. How many different hexagons can you draw thathave all sides equal?

b. How many different hexagons can you draw thathave all sides equal but not all angles equal?

c. How many different hexagons can you draw thathave exactly 2 right angles?

d. How many different hexagons can you draw thathave exactly 3 right angles?

e. How many different hexagons can you draw thathave exactly 4 right angles?

f. How many different hexagons can you draw thathave exactly 5 right angles?

g. Can you make a trapezoid with no obtuse angles?

h. How many different kinds of quadrilaterals can youmake that have exactly two opposite right angles?

i. Can you make a concave quadrilateral with exactly2 right angles?

j. Can you make a concave pentagon with exactly2 right angles?

16. Use a Venn diagram to represent the relationship between:

a. scalene and obtuse triangles.

b. equilateral and isosceles triangles.

c. parallelograms and rectangles.

d. rectangles, rhombi, and squares.

17. Write in the labels for each set in the problems below.Justify your choice. Add at least one new figure to one ofthe regions.

a.

b.

18. Show that the family tree for quadrilaterals also holdsfor the characteristics of the diagonals.

19. In this section, we found that we could name some quad-rilaterals in terms of their “ancestors.” For example, wecould say that a rhombus is a parallelogram with fourequal sides. What other quadrilaterals could we describein terms of their ancestors?

20. Draw a triangle. Find the midpoints of each side. Con-nect those points. What is the relationship between thenew triangle and the original triangle? Prove it.

21. Take a blank piece of paper. Fold it in half and then foldit in half again. Draw a scalene triangle and then cut itout. You should now have four congruent triangles. Foreach of the following questions, explain your reasoning.

a. Put these triangles together to make a large triangle.

b. Is there only one way or are there different ways?

c. How could you prove that the new triangle is actu-ally a triangle, as opposed to a figure that is almost atriangle?

d. Does the new large triangle have anything in com-mon with the original triangle?

22. What is the relationship between the number of sides in a polygon and the total number of diagonals in thatpolygon?

23. Prove that the sum of the interior angles of any polygonis equal to where n represents the number ofsides in the polygon.

24. Describe all possible combinations of angles in a quadri-lateral (for example, acute, acute, acute, obtuse). Brieflysummarize methods you used other than random trialand error.

25. Trace this circle following #26 onto a blank sheet of paper. Describe as many ways as you can for finding the centerof the circle. In each case, explain why the method works.

26. Trace the circle onto a blank sheet of paper. After cuttingout the circle and finding the center, fold the paper sothat the top point of the circle just touches the center ofthe circle. Fold the circle again so that another point onthe circle just touches the center and the two folds meetat a point. The two folds will be congruent. Fold the circle one more time so that your three folds will all becongruent. What kind of triangle has been created insidethe circle? Prove that you will get this kind of triangle bymaking these three folds.

180�n � 2�,


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27.A 3RIT is a figure made from 3 right isosceles triangles.On the left are two examples of RITs, and on the right aretwo figures that are not RITs.

a. Write a definition of 3RIT.

b. Find as many different 3RITs as you can. Sketch (ortape, or glue) them on a separate piece of paper.

c. Are the two 3RITs atthe right different ornot? Explain why you believe they are (orare not) different.

d. Find as many 4RITSs as you can.

28. Make as many different hexiamonds as you can. A hexi-amond is made by joining six equilateral triangles—when a side is connected, a whole side connects to awhole side. Briefly (in two or three sentences) describeyour method(s) for generating as many shapes as pos-sible. Describe any method(s) other than random trial and error; there are many different ways to be systematic.Cut your hexiamonds out and tape them to a piece ofpaper. (There are more than 9 and fewer than 16.)

29. Find the following points on a sheet of graph paper:

A B C D

E F G H

Now connect the points in order. What do you see?

30. Find the distance between the pairs of points:

a. and b. and

31. Find the midpoints of the line segments connecting thesepairs of points:

a. and b. and

32. a. Three vertices of a kite are , and What are the coordinates of the fourth vertex?

b. The two vertices that form the base of an isosceles tri-angle are and What are the coordinatesof the other vertex?

c. The coordinates of the endpoints of the hypotenuse ofa right triangle are and Find the other ver-tex. There are two possible solutions.

d. Three vertices of a parallelogram are andFind the fourth vertex. There are three possible

solutions.

e. A rectangle is oriented so that its sides form verticaland horizontal lines. Two coordinates of a rectangleare and Do you have enough informa-tion to determine the other two coordinates? If so,find them. If not, explain why not and describe whatinformation you would need (for example, a third co-ordinate, the length of one side, the relative locationof one of the points).

e. Make up a problem similar to the ones above andsolve it.

33. We can play a variation of a child’s game called “Whatam I?” that I will call “Where am I?”

a. I am a square. The intersection of my two diagonalslies at the point and the length of each of mysides is 6. My sides form horizontal and vertical lines.Where am I?

b. I am an isosceles triangle. The midpoint of my base isthe point my base forms a horizontal line, andmy vertex is at the point Oh, I almost forgotto tell you. I am upside down. Where am I?

c. I am a right, isosceles triangle. I have an area of50 square units. The coordinates of the vertex atwhich the two sides meet is and my sides formhorizontal and vertical lines. Do you have enough in-formation to determine the other two coordinates? Ifso, find them. If not, explain why not and describewhat information you would need.

d. Make up a problem similar to the ones above andsolve it.

34. Determine the coordinates ofthe vertices of tangram pieces ifthe bottom left-hand corner isat the origin and if the length ofeach side of the square is 8.

35. Determine the coordinates ofthe vertices of the followingquilt pattern.

�0, 0�

�7, �9�.�7, �2�,

�3, 3�,

�7, 6�.��3, �2�

�0, 6�.�4, 0�,�0, 0�,

�3, 1�.�7, 5�

�2, 3�.��5, 3�

�12, 8�.�9, 11��6, 8�

�7, 12��3, 4��4, 10��0, 4�

�10, 10��4, 7��5, 7��2, 3�

�3, �3��0, �3��0, 5��2, 5��2, �5��5, �5��5, 5��3, 5�

3 RITNOT3 RIT

NOT3 RIT3 RIT


304150_ch_08_02.qxd 1/16/04 6:07 AM Page 545

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